Integration: the Area Problem

The formula to find areas underneath curves derived from scratch

Tip: This isn't the place to ask a question because the teacher can't reply.

Key Questions

• Let us look at the definition of a definite integral below.

Definite Integral
$\setminus {\int}_{a}^{b} f \left(x\right) \mathrm{dx} = {\lim}_{n \to \infty} {\sum}_{i = 1}^{n} f \left(a + i \Delta x\right) \Delta x$,
where $\Delta x = \frac{b - a}{n}$.

If $f \left(x\right) \ge 0$, then the definition essentially is the limit of the sum of the areas of approximating rectangles, so, by design, the definite integral represents the area of the region under the graph of $f \left(x\right)$ above the x-axis.

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